Gravity is the force of inherent natural attraction between two massive bodies. The magnitude of the gravitational force is directly related to the mass of the bodies and is inversely related to the square of the distance between centers of mass of the two attracted bodies.
Gravity is measured as acceleration, g, usually as a vertical vector component. The freefall acceleration, g, of an object near the surface of the earth is given to a first approximation by the gravitational attraction of a point with the mass of the entire earth, Me, located at the center of the earth, a distance, Re, from the surface of the earth. This nominal gravity value, g=G×Me/Re2, is about 9.8 m/s2. Thus, the freefall acceleration due to gravity near the earth's surface of an object having a small mass compared to the mass of the earth is about 9.8 m/s2. The common unit of measurement for gravity is the “Galileo” (Gal), which is a unit of acceleration defined as 1 cm/s2. One Gal generally approximates 1/1000 (10−3) of the force of gravity at the earth's surface. An instrument used to measure gravity is called a “gravimeter.”
The most accurate gravimeters are absolute gravimeters. Interferometric absolute gravimeters usually use a freely falling test mass and a laser or single-frequency light beam which reflects from the freely falling test mass. The reflected light beam is combined with a reference light beam to develop interference fringes. Interference fringes are instances where the amplitude or intensity of the reflected and reference light beams add together to create increased intensity, separated by instances where the two beams cancel or create diminished intensity.
Fringes occur on a periodic basis depending upon the change in the optical path length of the reflected beam relative to the optical path length of the reference beam. One fringe occurs whenever the optical path difference between the reflected and reference beams changes by the wavelength of the light of the two beams. When an object that is part of the beam path moves, it typically changes the path length by twice the amount of physical movement because the physical movement changes both the entry and exit of the beam path. In this circumstance, a fringe typically occurs when the object moves by one-half of a wavelength. The fringes taken together as a set comprise a record of the distance that the freely falling body moves.
Because the path length of the reflected beam changes as it is reflected from the freely falling test mass, and because the freefall movement of the test mass is established by gravity, the occurrence and timing of the resulting interference fringes defines the characteristic of gravity. The use of optical fringe interferometry to measure gravity characteristics is well-known. U.S. Pat. No. 5,351,122 describes an example of an absolute gravity measuring instrument, called a “gravimeter.”
A gradient of gravity is the rate at which gravity changes in a certain direction and over a certain distance. A gravity gradient is therefore the change or first derivative of the gravity over distance. Near-field variations in gravity are caused by localized variations in the mass or density of at least one of the two attracted bodies. An instrument used to measure a gradient of gravity is called a “gradiometer.”
Although the gradient of gravity can be determined in any direction, the vertical gradient of gravity is useful in many practical applications. Vertical gravity gradients identify changes in density or mass of a particular material or geological structure. For example, gravity gradients are used to establish the location of underground geological structures, such as a pool of liquid petroleum encased within an earth formation, narrow seams or “tubes” of high density geological materials such as diamonds or cobalt, or voids in a geographical formation caused by a tunnel or cavern. These changes in the subterranean material density are most measurable within a relatively short near-field distance, typically within a few hundred meters.
Subsurface density anomalies, for example from valuable nearby high density ore bodies or voids caused by tunnels or areas of low density material, affect the local value of gravity, g, at a level of about 1 part per million ( 1/106), and in some cases 1 part per billion ( 1/109). The large background of the earth's gravity requires that any direct gravity measurement to detect such subsurface anomalies have a very large dynamic range of parts per billion, otherwise direct gravity measurements will not be useful for locating and detecting such subsurface density anomalies. It is difficult to make gravimeters with such levels of extremely high precision, so it is desirable to find ways to cancel the large effect of the earth's gravity while preserving the ability to detect gradations in nearby density anomalies.
The vertical gravity gradient of the earth is typically measured in terms of a unit called the Eotvos unit, E, given by 10−9/s2. The vertical gravity gradient of the entire earth is typically about 3000E. Typical nearby mass anomalies can affect the vertical gravity gradient at a level of about 1 E or more. Thus, the contrast of the vertical gravity gradient caused by nearby mass anomalies to the earth's vertical gravity gradient is about 300,000 (3×105) times larger than for the gravity value itself. This means that a vertical gravity gradiometer can have 3×105 times less precision than a gravimeter and still be used effectively to detect or locate nearby mass or density anomalies.
A gradiometer removes the effect of gravity. Logically, a gradiometer differences the gravity measurements at two different nearby locations. A known vertical gravity gradiometer is made by placing two gravimeters above one other with a vertical separation of fixed distance, z, and then subtracting the two gravity measurements, g1 and g2. The vertical gravity gradient, γ, is then given by the ratio of this difference divided by the vertical separation, i.e. γ=(g2−g1)/z. This quantity is also mathematically referred to as the spatial derivative of gravity in the vertical direction.
One or more absolute gravimeters can be used to measure the gravity at the different locations, typically one above the other. The gravity measurements are subtracted and then the result is divided by the distance between the locations of the two gravity measurements to obtain a gravity gradient measurement.
The separate gravity measurements can be obtained approximately simultaneously with multiple instruments or at separated time intervals with the same instrument if the gravity is not expected to change significantly between the times of the multiple measurements. The distance between the locations of these separate measurements is also measured. Each of these multiple separate measurements involves some risk and amount of error.
Each gravimeter used in measuring the gravity is also subject to naturally-occurring and man-made vibrations and other physical perturbations. These vibrations and perturbations cause minute changes in the path length of the reflected and reference light beams in a light beam interferometric instrument, causing interference fringes which are not related to the gravity characteristic measured. Such anomalous interference fringes reduce the accuracy of the measurement and enhance the potential for errors. Further still, each of the instruments is subject to unique vibrations and physical perturbations, which magnify the range of error when the measurements are subtracted from one another.
Attempts have been made to eliminate the anomalous vibration and perturbation errors through common mode rejection. In theory, connected-together instruments are subject to the same physical influences, thereby introducing the same error into all the measurements. When the measurements are subtracted, the common error in both signals is theoretically canceled or rejected. However, the practical effect falls substantially short of complete common mode rejection.
It is practically impossible to achieve a sufficiently rigid connection between the two instruments to cause both to experience the same degree of perturbation. It is impossible to freefall the test masses of the instruments at the same time, so each measurement is always subject to anomalies that do not influence the other measurement. The environments in which the test masses fall in the separate instruments are not the same, despite the attempt to create a vacuum around the test masses in the instruments. The vacuum surrounding each test mass has a slightly different amount of residual gas which creates a slightly different aerodynamic drag on each freefalling test mass. The different amounts of aerodynamic drag influence the freefall characteristics of each test mass differently, thereby introducing discrepancies. Further still, the optics which conduct the light beams in the connected instruments are slightly different, and those differences introduce unique discrepancies. Even slight changes in temperature or pressure may affect the optics of each instrument differently. Physical movement caused by vibration or perturbation of the external optical fibers or elements which conduct the input and output light beams into and from each instrument introduce unique phase shifts, which also influence the measurements. Separate laser light sources for each instrument create unique phase changes in the light beams, which introduce anomalous fringe effects that may introduce measurement errors. Inadvertent slight angular rotation or tilting of one or both the test masses during simultaneous freefall changes the length of the reflected light paths in that instrument, which again contributes to error when the two gravity measurements are subtracted to determine the differential gradient of gravity.
These and other unique and adverse influences increase the possibility of deriving inaccurate measurements. In addition, the mathematical manipulations of subtracting the measurements and dividing by the distance between the measurement locations may compound the errors. These and other errors are not subject to common mode rejection, because the errors uniquely affect some singular aspect of one instrument and not any other instrument used. The inability to achieve effective common mode rejection makes the measurement of a gradient of gravity using gravimeters error-prone, particularly in vibration-prone or perturbation-prone environments.